Specification Error

Specification Error

When constructing any regression model, we are always most interested in explaining what variables cause the dependent variable to change and by how much. This will always depend on a combination of economic theory; basic human behavior; and past experience.

Nội dung Text: Specification Error

Nguyeãn Troïng Hoaøi Analytical Methods 9 1
Specification Error
When constructing any regression model, we are always most interested in
explaining what variables cause the dependent variable to change and by how
much. This will always depend on a combination of economic theory; basic human
behavior; and past experience.
One of the assumptions of OLS is that the model is correctly specified. The
specification error can be explained by these two aspects : -
a) Missing / omitting relevant information / explanatory variables or from
including irrelevant variables.
b) Incorrect functional form.
This lecture will discuss the following issues : which regressors should be included
and / or excluded from a particular model. In other words, we will consider the
following cases : -
a) A regression model that excludes some important explanatory variables.
b) A regression model that includes some irrelevant regressors.
1) Exclusion of relevant variables
Suppose that we are interested in the following model : -
Yi = β1 + β 2 X 2i + L + β K X Ki + β ( K + 1 ) X ( K +1) i
+ L + β( K + L ) X ( K + L ) i + εi
The question is whether the set of L regressors - X( K + 1 ) + L + X( K + L ) - are
important variables that should be included in the model.
But because of a certain reason, we have to use the following model : -
Yi = β1 + β 2 X 2i + L + β K X Ki + ε i
For illustration, we can use a model with only two explanatory variables. The
model with two explanatory variables is specified as follows : -
True model Yi = β1 + β 2 X 2i + β 3 X 3i + ε i 9.1
Note: we assumed that X2 and X3 are the two important regressors that explain the dependent
variable Y, that is, we expect that β 3 # 0. The model we use to estimate is as follows : -
Estimation model Yi = β1 + β 2 X 2i + ε i 9.2
This means we have excluded an important regressor X3i.
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This regression shows that HOUSING is explained quite well through GNP and
INT.RATE. If we temporarily assume that this is the true model, we then regress
HOUSING against GNP.
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We can conclude that this model excluded an important explanatory variable -
INT.RATE (Observe how the coefficient of determination; the coefficient of GNP;
and the standard error of the estimator of GNP change).
Conduct another regression : INT.RATE on GNP
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∧
Therefore, the standard error of the estimator β 2 will be inaccurate (unstable, or
biased), and thus the use of its standard error is inaccurate, too. As a result, any
hypotheses testing will be invalid. From looking at the regression results, we will
easily see that.
For caution, we use the Wald test for a restricted model (an estimated model) and
for an unrestricted model (a true model), based on the hypothesis that β 3 = 0.
2. Including irrelevant variables
To analyze this case, we return again to the two-regressor model, only this time
we assume that X3 does not relate to Y (that is β 3 = 0 ). In other words, X3 is
irrelevant.
True model Yi = β1 + β 2 X 2i + ε i
Estimated model Yi = β1 + β 2 X 2i + β 3 X 3i + ε i
The estimated model has the following criteria : -
a) Estimators of other coefficients (except X3) are unbiased and consistent.
Again, if we take the estimated coefficients and calculate their expectations : -
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For example, for including irrelevant variables in the equation, we can add two
more, such as population - POP - and unemployment - UNEMP - into the model : -
Now examine the regression results, especially for the two new variables.
Since we assume that the two new variables are irrelevant, we are going to do the
Wald test on these.
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3) General – to – Simple Modeling Strategy
The results that we have just established suggest that the general-to-simple
modeling strategy is superior to the simple-to-general strategy. The steps are as
follows : -
[ Use economic theory, previous research, and experience to specify a
general model (in this case “general” means a model that includes all
possible relevant regressors).
[ Estimate the model
[ If any of the coefficients are statistically insignificant, omit the least
significant one and re-estimate. Variables are eliminated one-by-one
because of the effect of the elimination on the remaining variables. If the
first regression shows two insignificant variables, and the least significant
one is then omitted, this may increase the significance of the remaining
one.
[ From using the Wald Tests to test the final model (the restricted model),
compare against the initial general model (the unrestricted model).
4) An application of modelling Strategy
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